Scalar/vector projections | Year 11

01 / LEARN

The idea, step by step

The part along a direction

The scalar projection of u onto a non-zero direction vector v is the signed component of u in that direction, in the same units as u. It is positive when u points partly in the same direction and negative when it points partly against it.

The vector projection is this signed scalar multiplied by the unit vector along v. Its result is a vector parallel to v. The direction vector need not itself be a unit vector; the formulas account for its length.

{comp}_{v} u = \frac{u \cdot v}{∣ v ∣}, {proj}_{v} u = \frac{u \cdot v}{v \cdot v} v

The leftover is perpendicular

Subtracting the projected vector from u leaves a perpendicular component. Dotting this remainder with v gives zero. This provides a direct check of a projection calculation.

Projection is directional: projecting u onto v is generally different from projecting v onto u. Never divide by the length of the vector being projected when the formula requires the direction vector's length.

A common trap

Treating the scalar projection as a non-negative length in every case.

Instead: A scalar projection is signed. The length of the projected vector is the absolute value of that scalar.

02 / WORKED EXAMPLES

See the thinking

Example 1

Projection on a diagonal direction

Find the scalar and vector projections of u=(4,2) onto v=(1,1).

$u \cdot v = 6$ and $∣ v ∣ = \sqrt{2}$ .
The scalar projection is $6 / \sqrt{2} = 3 \sqrt{2}$ .
The vector projection is $(6 / 2) (1, 1) = (3, 3)$ .

Scalar

3 \sqrt{2}

; vector

(3, 3)

Example 2

A negative component

Find the projection of a=(-3,4) onto b=(1,0), then its perpendicular remainder.

The scalar projection is $- 3$ .
The vector projection is $(- 3, 0)$ .
Subtracting gives $(- 3, 4) - (- 3, 0) = (0, 4)$ .

Parallel component

(- 3, 0)

; perpendicular component

(0, 4)

Check your understanding

Project $(5, 2)$ onto the y-axis direction $(0, 1)$ .

Show answer

$(0, 2)$ .

The dot product with $(0, 1)$ is $2$ , and that direction is already a unit vector.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the scalar projection of $(3, 4)$ onto $(1, 0)$ .
2 marks
Find the vector projection of $(2, 5)$ onto $(0, 2)$ .
2 marks
Project $(5, 1)$ onto $(1, 1)$ .
2 marks
Resolve $(4, 2)$ into components parallel and perpendicular to $(1, 1)$ .
3 marks
Find the scalar projection of $(- 2, 3)$ onto $(2, 0)$ and the length of the projected vector.
3 marks
Show that replacing v by $2 v$ does not change ${proj}_{v} u$ .
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the scalar projection of $(3, 4)$ onto $(1, 0)$ .
2 marks
The direction has length one and the dot product is three.
$3$ .
Find the vector projection of $(2, 5)$ onto $(0, 2)$ .
2 marks
Coefficient $= (10 / 4) = 5 / 2$ .
Multiply by $(0, 2)$ .
$(0, 5)$ .
Project $(5, 1)$ onto $(1, 1)$ .
2 marks
The dot product is six and squared direction length is two.
$(3, 3)$ .
Resolve $(4, 2)$ into components parallel and perpendicular to $(1, 1)$ .
3 marks
Calculate the projection, then subtract it.
Check $(1, - 1) \cdot (1, 1) = 0$ .
Parallel $(3, 3)$ ; perpendicular $(1, - 1)$ .
Find the scalar projection of $(- 2, 3)$ onto $(2, 0)$ and the length of the projected vector.
3 marks
The dot product is $- 4$ , divided by direction length $2$ .
Take the absolute value only for the length.
Scalar $- 2$ ; length $2$ .
Show that replacing v by $2 v$ does not change ${proj}_{v} u$ .
4 marks
The numerator doubles, the denominator quadruples and the final direction vector doubles.
The factors cancel.
Both give $\frac{u \cdot v}{v \cdot v} v$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: vectors in the plane. Covers magnitude, components, geometry, scalar products, projections and proofs; follows the approved Specialist preparation structure.

Official curriculum source ↗

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